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Square roots of elliptic second order divergence operators on strongly Lipschitz domains: \(L^p\) theory. (English) Zbl 1161.35350
Summary: We study \(L^p\) estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form \(L=-\text{div}(A\nabla)\)on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our earlier work [Square root problem for divergence operators and related topics. Astérisque. 249. Paris: SMF (1998; Zbl 0909.35001)] where we considered operators on \(\mathbb R^n\). We obtain among other things \(\| L^{1/2}f|_p\leq c\| \nabla f\|_p\) for all \(1<p<\infty\) if \(L\) is real symmetric and the domain bounded, which is new for \(1<p<2\). We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary.

35J15 Second-order elliptic equations
47A60 Functional calculus for linear operators
42B25 Maximal functions, Littlewood-Paley theory
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